A general form of twisted Hermite Gaussian Schell-model (THGSM) beams is introduced; analytical expressionsare obtained for cross-spectral density and M2-factor using the extended Huygens-Fresnel principle and Wigner function. The evolution of THGSM beams during propagation in non-Kolmogorov turbulence is shown numerically; the beams exhibit self-splitting and twist into two lobes. The intensity distribution evolves into a Gaussian shape and beam quality worsens with increasing distance; the intensity distribution and M2-factor are determined by the twist factor, beam orders, and other beam parameters. THGSM beams provide more degrees of freedom to regulate beam parameters, thereby enriching the types of partially coherent beams.